# Identities for the Multiple Polylogarithm Using the Shuffle Operation

2001

## Level of Access Assigned by Author

Open-Access Thesis

## Degree Name

Master of Arts (MA)

Mathematics

David M. Bradley

Henrik Bresinsky

William Snyder

## Abstract

At the beginning of my research, I understood the shuffle operation and iterated integrals to make a new proof-method (called a combinatorial method). As a first work, I proved an combinatorial identity 2 using a combinatorial method. While proving it, I got four identities and showed that one of them is equal to an analytic identity 1 which is found at the paper [2] written by David M. Bradley and Doug Bowman. Furthermore, I derived an formula involving nested harmonic sums. Using Maple (a mathematical software), I found a new combinatorial identity 3 and derived two formulas: One is related to multiple polylogarithms and the other is related to rational functions. Since letters in the identities represent differential 1-forms which converge, I can find new formulas - - - if I get a proper setting. - My research was developed by considering a combinatorial identity 4 given by David M. Bradley, thesis advisor. Though it looked very complicated, the implication for the identity was very interesting to me. Using a combinatorial proof-method, I proved it. Even though I just derived one formula involving nested harmonic sums in this thesis, the identity has potentiality because, if I 6nd a new setting for differential 1-forms, I can derive a new formula involving multiple polylogarithms. It was not very easy to prove the combinatorial identity 4 even though I used the combinatorial proof-method as I did at the proofs of the combinatorial identity 2 and 3. The reason is that the result of the identity 4 is more complicated than those of the identities 3 and 4. So, Lemma 5 is needed to complete the proof of the identity 4, which step is not needed in the proofs of the identities 3 and 4. When formulating the identity 4, I had a trouble in defining the notations because of their complexity. When I formulated the identity 4, it was a beautiful formula. As we can see in the paper [3], there are various conjectures related to multiple zeta values whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. At this situation, this combinatorial proof-method can play a crucial role in developing other fields such as knot theory and quantum field theory as well as combinatorics.